NDLI: Graph Embeddings and Laplacian Eigenvalues

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Miller, Gary L. Guattery, Stephen
Copyright Year	2000
Abstract	Graph embeddings are useful in bounding the smallest nontrivial eigenvalues of Laplacian matrices from below. For an n n Laplacian, these embedding methods can be characterized as follows: The lower bound is based on a clique embedding into the underlying graph of the Laplacian. An embedding can be represented by a matrix $\Gamma$; the best possible bound based on this embedding is $n/\lambda_{\max} (\Gamma^T \Gamma)$, where $\lambda_{\max}$ indicates the largest eigenvalue of the specified matrix. However, the best bounds produced by embedding techniques are not tight; they can be off by a factor proportional to log2n for some Laplacians.We show that this gap is a result of the representation of the embedding: By including edge directions in the embedding matrix representation $\Gamma$, it is possible to find an embedding such that $\Gamma^T \Gamma$ has eigenvalues that can be put into a one-to-one correspondence with the eigenvalues of the Laplacian. Specifically, if $\lambda$ is a nonzero eigenvalue of either matrix, then $n / \lambda$ is an eigenvalue of the other. Simple transformations map the corresponding eigenvectors to each other. The embedding that produces these correspondences has a simple description in electrical terms if the underlying graph of the Laplacian is viewed as a resistive circuit. We also show that a similar technique works for star embeddings when the Laplacian has a zero Dirichlet boundary condition, though the related eigenvalues in this case are reciprocals of each other. In the zero Dirichlet boundary case, the embedding matrix $\Gamma$ can be used to construct the inverse of the Laplacian. Finally, we connect our results with previous techniques for producing bounds and provide an example.
Starting Page	703
Ending Page	723
Page Count	21
File Format	PDF
ISSN	08954798
DOI	10.1137/S0895479897329825
e-ISSN	10957162
Journal	SIAM Journal on Matrix Analysis and Applications (SJMAEL)
Issue Number	3
Volume Number	21
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-31
Access Restriction	Subscribed
Subject Keyword	Eigenvalues, singular values, and eigenvectors graph embeddings graph eigenvalues and eigenvectors Graphs and linear algebra Laplacian matrices Inequalities involving eigenvalues and eigenvectors
Content Type	Text
Resource Type	Article
Subject	Analysis

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